This is a complex situation
As Dr Timmerman says above, there's a lot going on here. In my experience, it's almost never a good idea to file down canines, as they are the cornerstone of you bite, regardless of what philosophy a well trained dentist subscribes to. Obviously I don't know the whole story here, but I agree that you should dot all your i's and cross all your t's before pursuing any treatment. Find a top notch orthodontist to be sure the breathing issues are addressed. Best of luck to you!
I can clearly see from your photo that you have a narrow upper archa and a few of your teeth are in cross bite. If you first expand your upper arch with Invisalign then your front teeth will be allowed to move down on there own because the molars are seating together deeper--it's hard to explain over the internet.
Also your dentist can use attachments and elastics with Invisalign if he knows what he's doing, where you wouldn't need brackets at all. The other answers to your question trouble me if they think you need combination treatment.
30% of those with Invisalign had braces before
You look like one that has trouble breathing through his mouth. When that happens the tongue drops down automatically to narrow the upper jaw and open the bite. You have a hard time biting small things in the front.
Your tongue needs to be trained to touch the roof of your mouth as you rest. The upper jaw should be the shape of your tongue. Yours is not. The upper needs to be expanded. As the jaw narrowed your teeth then did not have room to remain straight.
Invisalign can widen your teeth and close your bite. Then plan on wearing a retainer every night the rest of your life so you don't face the same situation a third time.
Crossbite and airway issues should be addressed when treatment planning braces
Teeth fit together like gears and when they don't it can create an "end to end" relationship (and distance the nose and chin). When this is corrected, some of the front teeth can appear to "come down" more. But the cause should be addressed. Sometimes there is a tongue positioning habit, and if not fixed it will push everything right back.
There should be an airway evaluation as well (which could create the tongue positioning habit). This could help avoid serious sleep apnea issues later in life.
The upper arches look narrow and may need expansion.
Complex cases like this should be thoroughly researched. I would suggest a few consultations to find a plan you are comfortable with.
It sounds like the treating dentist has a plan from beginning to end. You have had orthodontics before and your teeth relapsed to where they really wanted to be. Treating the malocclusion may disrupt the balance that your body has naturally found by putting the teeth back into a crooked position.
The picture is blurry, but it seems like one of your central incisors is discolored. You may be able to fix the color on that and do 4 veneers on the front teeth and get a reasonable straightness without having your teeth put back out of balance and need retainers. That would be an option you should discuss with a dentist.
Second and third opinions never hurt, and I in fact, encourage it. Go see a cosmetic dentist and perhaps have them mock-up the teeth so you can visually see them, and see an orthodontist and you will have done some good due dilligence.
Please see an orthodontist
I would recommend seeing an orthodontist who can give you different treatment options as it seems like you have seen a general dentist. An orthodontist is a specialist that has 2-3 years of additional training and specializing in treating malocclusions, I have not seen your full records, but it would seem to me that you should do either fixed braces with elastics to align the teeth and close the open bite OR invisalign with elastics to align the teeth and close the bite.
Crowded Teeth and Open Bite in Front
It sounds like your dentist is going to do a combination of metal brackets and Invisalign Express. This may be a good plan for you. If this is, in fact, the treatment plan,my only concern would be that the Invisalign Express does not allow for refinements. So it would be more difficult to do the finishing touches at the end of the case. For your bite (filing down the canines), make sure your dentist has specific training in this area. When I alter someone's bite, I have to make sure I'm keeping the jaw joints aligned and keeping all forces distributed properly.